Root graded groups of type $ H_3 $ and $ H_4 $
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911035573665792 |
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| author | Berg, Lennart Wiedemann, Torben |
| author_facet | Berg, Lennart Wiedemann, Torben |
| contents | Using the well-known realisation of the root system $ H_4 $ as a folding of $ E_8 $, one can construct examples of $ H_4 $-graded groups from Chevalley groups of type $ E_8 $. Such Chevalley groups are defined over a commutative ring $ R $, and the root groups of the resulting $ H_4 $-grading are coordinatised by $ R \times R $. We show that every $ H_4 $-graded group arises as the folding of an $ E_8 $-graded group, or in other words, that it is coordinatised by $ R \times R $ for some commutative ring $ R $. We also prove similar assertions for $ (D_6, H_3) $ in place of $ (E_8, H_4) $. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_06745 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Root graded groups of type $ H_3 $ and $ H_4 $ Berg, Lennart Wiedemann, Torben Group Theory Rings and Algebras 20E42 (Primary) 17C05, 20F40, 51C05 (Secondary) Using the well-known realisation of the root system $ H_4 $ as a folding of $ E_8 $, one can construct examples of $ H_4 $-graded groups from Chevalley groups of type $ E_8 $. Such Chevalley groups are defined over a commutative ring $ R $, and the root groups of the resulting $ H_4 $-grading are coordinatised by $ R \times R $. We show that every $ H_4 $-graded group arises as the folding of an $ E_8 $-graded group, or in other words, that it is coordinatised by $ R \times R $ for some commutative ring $ R $. We also prove similar assertions for $ (D_6, H_3) $ in place of $ (E_8, H_4) $. |
| title | Root graded groups of type $ H_3 $ and $ H_4 $ |
| topic | Group Theory Rings and Algebras 20E42 (Primary) 17C05, 20F40, 51C05 (Secondary) |
| url | https://arxiv.org/abs/2408.06745 |